Applications of an alternative formulation for one-layer real time optimization

APPLICATIONS OF AN ALTERNATIVE FORMULATION FOR ONE-LAYER REAL TIME OPTIMIZATION

A.L.Schiavon Júnior1^* and R.G.Corrêa2

¹Universidade de São Paulo, EPUSP, São Paulo - SP, Brasil

²Universidade Federal de São Carlos, UFSCar, São Carlos - SP, Brasil

E-mail: [e-mail], [e-mail] ,

[e-mail]

(Received: November 23, 1999 ; Accepted: April 6, 2000)

Abstract - This paper presents two applications of an alternative formulation for one-layer real time structure for control and optimization. This new formulation have arisen from predictive controller QDMC (Quadratic Dynamic Matrix Control), a type of predictive control (Model Predictive Control – MPC). At each sampling time, the values of the outputs of process are fed into the optimization-control structure which supplies the new values of the manipulated variables already considering the best conditions of process. The variables of optimization are both set-point changes and control actions. The future stationary outputs and the future stationary control actions have both a different formulation of conventional one-layer structure and they are calculated from the inverse gain matrix of the process. This alternative formulation generates a convex problem, which can be solved by less sophisticated optimization algorithms. Linear and nonlinear economic objective functions were considered. The proposed approach was applied to two linear models, one SISO (single-input/single output) and the other MIMO (multiple-input/multiple-output). The results showed an excellent performance.

Keywords: Real Time Optimization, Predictive Control,

INTRODUCTION

A way for introducing the optimization in the control of processes is to use a hierarchical structure denominated multi-layer structure (Kwong, 1992) that can be represented in its simplified form in Figure 1.

The optimization layer tries to find (based an economical model) the best operational conditions of the process. The regulation layer executes the actions of the regulatory control. The layer of process is the layer of the instrumentation and of unitary operations that will be represented by simplified models.

In the last two decades it has been very important the development of control techniques that uses a model of the process as the controller’s part, denominated Model Predictive Control (MPC). The controller occupies the regulation position inside the multi-layer structure. The formulation of this class of controllers have allowed a larger interaction between the control and optimization layers due to its prediction characteristic.

An interesting question concerning on optimization and control is the following: which is the best form of interaction between the optimization layer and the regulatory control layer? Several forms of interaction exist:

(a) the optimization is performed off-line using a computer and the set-points are implemented by the operator into the plant. Latour (1979) has already said that this form of interaction does not use all the potential benefits of on-line steady state optimization. Open-loop operator control might achieve some fraction of automatic closed-loop optimization;

(b) the stationary optimization is performed on-line in an interval larger than the interval of the regulatory actions (Kwong, 1992). That is, when the plant set-points are altered, it is awaited the plant to reach a new stationary state to calculate new values of the set-points. The problem is that for a very disturbed plant, the set-points may be rarely corrected (the optimization layer has little actuation);

(c) the optimum set-points are calculated in the optimization level and then sent to an advanced control strategy (normally a predictive controller) located in the control level. The optimization and the control actions are executed at the same frequency, however in a two-layer structure with strong interaction between them (Moro & Odloak, 1995; Schiavon, 1998). Since the control actions and optimization are executed in the same frequency, the plant operates most of the time close to the best operational conditions;

(d) the optimization and the control problems are solved together in an one-layer structure. The economical optimization problem is directly included in the predictive controller (Yousfi & Tournier, 1991; Gouvêa & Odloak, 1998; Schiavon, 1998). This structure also allows the plant to operate most of the time close to the best operational conditions and it still provides a computational gain for using an only stage (algorithm) to execute the control actions and optimization. Another advantage is the possibility of using nonlinear economical functions;

In the first and second structures the actions of the optimization and regulation layers are completely independent. In the third structure a strong interaction exists among them but their actions are accomplished in different procedures. In the last structure the optimization and control layers forming an only layer that calculates the control actions taking into account the best operational conditions of the process.

The last two interaction forms have become nowadays the most investigated real time optimization (RTO) strategies. In the two-layer optimization structure proposed by Moro & Odloak (1995), the regulatory control layer is a conventional unconstrained multivariable Dynamic Matrix Control (DMC) and the optimization layer solves a linear programming (LP) problem based on the DMC formulation. The DMC defines the future (or predicted) steady state of the process and the optimization layer includes all process constraints. Instabilities in the operation with such interaction form may occur if the process is disturbed at the same time in which the set-points are changed, unless the tuning of the controller is made conservative (Gouvêa & Odloak, 1998). In the two-layer optimization structure proposed by Schiavon (1998) the regulatory control layer is based on a constrained QDMC and the optimization layer solves a Nonlinear Programming (NLP) problem.

In the beginning of the nineties, Yousfi & Tournier (1991) proposed an one-layer real time optimization structure using predictive control. In the paper of Yousfi & Tournier, a steady state optimization is incorporated into a Model Predictive Control (MPC). To introduce the optimization into the regulatory control, these authors added an economical term into the objective function to SMOC (Shell Multivariable Optimizing Control). Due to the linear nature of the economical problem (linear model and linear objective functions) considered by Yousfi & Tournier (1991) this approach is not normally suitable since most of the real problems are nonlinear. Odloak & Gouvêa (1996) have considered a similar approach but they used a linearization of the economical objective function. This approximation in the DMC controller may make the closed-loop system unstable. The inclusion of an economical nonlinear optimization problem directly into the DMC was implemented by Gouvêa & Odloak (1998) in a FCC (Fluid Catalytic Cracking) unit model. In such approach, a nonlinear programming (NLP) problem must be solved, requiring a robust and efficient solver that should guarantee the process under control. The authors compared the two-layer structure performance and the one-layer structure. They observed that the one-layer structure presented a better performance. Other approach to one layer control-optimization was development by Schiavon (1998), whose development is presented in this paper.

In this alternative formulation for the one-layer structure, the inverse of the process gain is used into QDMC. Then, a convex problem is obtained which can be solved with less sophisticated optimization algorithms. The performance of the controller is analyzed when linear and nonlinear economic objective functions are used. This approach was applied to two linear models, one SISO and another MIMO.

SISO CASE – LEVEL TANK

A schematic representation of the level tank is showed in Figure 2.

The SISO linear model of a level tank is

where Fs is proportional to h (tank level):

Fs = kh

(2)

Fi2=unknown disturbance (m³/min),

Fi1=manipulated variable (m³/min),

h=output variable (m),

k=1m²/min,

A=10m². Initial steady state data for the level tank is:

Fi1=5 m³/min,

Fi2=5 m³/min,

Fs=10 m³/min and h=10m.

Table 1 shows the bounds imposed on the optimization-control problem.

MIMO CASE

The MIMO case to be studied it is composed by two manipulated inputs (u₁ and u₂), two controlled outputs (y₁ and y₂) and a disturbance (u₃).

In Figure 3 is showed the diagram of blocks with transfer functions that represents the MIMO case:

The MIMO model can also be represented by the following transfer function matrix equation:

Bound constrains are applied to controlled and manipulated variables.

Table 2 shows the bounds imposed on the optimization-control problem for the MIMO case.

QDMC CONTROLLER

Among the controllers of the type MPC is QDMC (Quadratic Dynamic Matrix Control) that solves an optimization problem, calculating l (control horizon) control movements (Du) from the minimization of the sum of r (prediction horizon) values futures of the square of the difference among the set-points and the outputs calculated by the model. Only the first control action is implemented. The prediction is made from the convolution model. The convolution model calculates the predicted outputs from the current output and n (horizon of the model) past values of the inputs. The algorithm QDMC represents only the layer of regulatory control inside the multi-layer structure, not considering the optimization layer of process. A simplified formulation of this controller is showed below.

The values of the manipulated variables are calculated of the following NLP problem:

(4)

A is the dynamical matrix of the process model. E’ is the vector of predicted error considering only the past actions of control and (–ADu + E) is the vector of predicted error already considering the future control actions.

The term (–ADu + E)^TQ^TQ(–ADu + E) represents the sum of r values of the difference squares between the set-points and the outputs. Q (dimension rxr) is the weight matrix of the predicted error vector. The term (Du^TRDu) represents a weight in the l future control actions. R (dimension lxl) is the weight matrix of the control movements. Notice that the independent variable to be calculated in the NLP is only Du.

The minimization (eq. 4) is subject the following inequality constraints:

(5)

(6)

(7)

Larger details regarding the QDMC can be found in specific papers about predictive controllers (MPC). Basic aspects can be seen in Marchetti et al (1983), Ricker (1985), Garcia et al (1989) and Schiavon (1998).

ONE-LAYER OPTIMIZATION STRUCTURE

The one-layer optimization-control structure presented in this paper is based on the approach of Odloak & Gouvêa (1996) and Gouvêa & Odloak (1998) with a few but important modifications. The similarities among all of them are (1) the optimization and the regulatory control are performed at the same frequency and they are solved together; (2) the future predicted outputs are supplied by the MPC. The main differences are: in this work (1) the future stationary outputs and the future stationary control actions are calculated from the matrix of the inverse gain while in another are calculated directly from the prediction model and sum of control movements, respectively; (2) in this work a convex problem is generated, which can be solved with less sophisticated optimization algorithms while in the another it is necessary to use an algorithm that solves a non-convex problems.

The simulation of the one-layer strategy can be represented by the scheme on Figure 4. Observe that the regulatory control and the optimization form a single optimization problem.

FORMULATION OF THE NEW STRATEGY

In this formulation, at each time interval, the new values of the manipulated variables are supplied by the optimization-control structure based on the optimization of the stationary conditions of process. The movements of control and set-points are calculated by the following NLP problem:

(8)

Subject to the equality constraints:

(9)

(10)

(11)

and subject to the following inequality constraints:

(12)

(13)

(14)

Notice that there are three terms weighted in the Equation (8). The first two are the same ones used in the algorithm QDMC, meantime with weights w₁ and w₂. The third term (w₃(F)), constituted of the economic function weighted by w3, is responsible for the stationary optimization of the process. All three terms are now the objective function of the one-layer real time optimization structure. The bar on top of y and u in the economical function (eq. 9, 10 and 11) indicate that the outputs and inputs calculated take into account the alteration of the set-point. Observe that, different from QDMC, the independent variables calculated in NLP are Du and Dy^set.

Equations (10) and (11) calculate, at each interval, the future steady state and the future manipulated variables, respectively, to reach this future steady state based on the continuous optimization of the operational conditions of the process. This alternative formulation is based on the use of the inverse of the process gain (A_n).

A convex problem is generated when the constraints are represented by Equations (12) to (14). Observe that no constraints are imposed on the predicted outputs, but only on the set-points. In this situation the constraints do not change and the search region will be always convex. There are no crossing of constraints that can eliminate the feasibility region.

It was used the function constr (Optimization Toolbox) of MATLAB in the calculation of the control and set-points movements. This function uses an algorithm denominated Successive Quadratic Programming (SQP).

SIMULATION RESULTS

In the simulation were studied two situations. The first one is when a linear economical objective function (F) is used and the second one is when a nonlinear objective function is considered.

Simulations results of SISO CASE.

The parameters used in this case are shown in Table 3 and they are chosen based on the rules presented by Seborg et al. (1989).

Linear economical objective function

This section considers the case in which the economical objective function is given by the linear function, F = :

Figure 5 shows that the set-point was moved to the optimal operating point almost immediately. This new value of set-point is the lower bound of h. The regulatory control part of one-layer structure appropriately rejects the disturbance.

Nonlinear economical objective function

This section considers the case in which the economical objective function is given by the following nonlinear function,

Notice in Figure 6 that the set-point has moved smoothly to the optimal value and Fi₁ has moved in the direction of eliminating the error between output and set-point. Notice that the error between the output and the value of the set-point is very small for the whole time. The output accompanies closely the best operational conditions of the process.

Simulations results of MIMO CASE

The adopted parameters are shown in Table 4 and they are also chosen based on the rules presented by Seborg et al. (1989).

Linear economical objective function

This section considers the case in which the economical objective function is given by the linear function, :

Figure 7 shows that the set-points have moved to the optimal operating points almost immediately. These new values of set-points are the upper and lower bounds of y₁ and y₂, respectively. The regulatory control part of one-layer structure appropriately rejects the disturbance.

Nonlinear economical objective function

This section considers the case in which the economical objective function is given by the nonlinear function, :

Figure 8 shows that the set-points have moved to the optimal values and u₁ and u₂ have moved in the direction of eliminating the error between outputs and set-points.

CONCLUSIONS

This paper studied applications of a new formulation for one-layer real time optimization. The presented formulation supports linear and nonlinear economical objective functions. Due to the use of the matrix of inverse gains there was not need for specific algorithms to treat non-convex problems resulting of quadratic programming. It was verified that the objective function parameters (w1, w2 and w3) were easily tuned. In Gouvêa and Odloak (1998) and Odloak and Gouvêa (1996) was used the structure in an one-layer and they had difficulties to adjust the parameters w1, w2 and w3 of the their formulation, what hasn’t occurred with this new formulation. They also said that it would be more appropriate an optimization algorithm that treated of non-convex problems to be used in the one-layer structure. In the new formulation the problem is always convex, which is a very significant advantage because the algorithms that treat of non-convex problems usually have an computational effort very larger. The possibility of using nonlinear functions in the economical model turns this more applicable structure to real problems.

The simulation results support these arguments, showing that the proposed approach performed satisfactorily.

ACKNOWLEDGMENTS

Support for this work was provided by Coordenadoria de Aperfeiçoamento de Pessoal de Ensino Superior – CAPES.

NOMENCLATURE

A Dynamic matrix or transversal area tank A_n Matrix of the stationary gains E’ Predicted error vector considering only the past actions of control f Feed flow, m³/min Fi₁ Manipulated variable, m³/min Fi₂ Disturbance variable in level tank, m³/min Fs Output flow, m³/min h Tank level, m I Identity matrix l Control horizon n Model horizon P_n Vector of the steady state effect of all past control actions Q Weighting matrix for the predicted errors r Optimization (or prediction) horizon R Weighting matrix for the control movements s Variable of Laplace space t Time T Sampling period, min u Manipulated variables and disturbances Manipulated variable considering alteration of the set-point y Controlled output Controlled variable considering alteration of the set-point y^set Set-point value y_k+n Future steady state value w_1,2,3 Weighting factors in the one-layer structure

Greek Simbols

F Economical objective function Du Control movements Dy^set Set-point movements

Subscripts

k Present time max Upper value min Lower value n Future steady state value

REFERENCES

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• Moro, L.F.L. & Odloak, D. Constrained multivariable control of fluid catalytic cracking converters. Journal of Process Control, 5, No. 1, p. 29 (1995).
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• Ricker, N. L. Use of Quadratic Programming for Constrained Internal Model Contro. Ind. Eng. Chem. Proc. Des. Dev., 25, p. 925 (1985).
• Schiavon Jr., A. L. Study of Two Real Time Optimization Structures with QDMC. M. Sc. Thesis, Universidade Federal de Sao Carlos, Sao Carlos (1998) (in Portuguese).
• Seborg, D.E., Edgar, T.F. and Mellichamp, D.A. Process Dynamics and Control. John Wiley & Sons, New York (1989).
• Yousfi, C. & Tournier, R. Steady state optimization inside model predictive control. Proc. ACC, p.1866 (1991).

*

To whom correspondence should be addressed

Publication Dates

• Publication in this collection
  16 Mar 2001
• Date of issue
  Dec 2000

History

• Received
  23 Nov 1999
• Accepted
  06 Apr 2000